Digital differential analyzers



Aug. 18, 1959 R. v. BENAGLlo ET AL 2,900,135

DIGITAL DIFFERENTIAL ANLYZEBS 6 Sheets-Sheet l Filed June 18, 1953 Aug.18, 1959 R. v. BENAGLIO ET Al.

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1N VEN TORS @ma 14 [5f/w ma .mc/f M. PANE/asm BY HAM 5 A. P/Pf/e MR.@KJR United States Patent DIGrrAL DIFFERENTIAL ANALYzERs Reno V.Benaglio, Birmingham, Jack M. Patterson, Berkley, and Charles A. Piper,Detroit, Mich., assignors to Bendix Aviation Corporation, Detroit,Mich., a corporation of Delaware Application June 18, 1953, Serial No.362,584

Claims. (Cl. 23S-152) This invention relates to digital differentialanalyzers` and more particularly lto a system for enhancing the ac--curacy in the operation of digital dilerential analyzers in obtainingthe solution of mathematical problems.

In co-pending application Serial No. 217,478 tiled March 26, 1951, byFloyd G. Steele and William F. Collison, a digital differential analyzeris disclosed for solving complex differential equations by digitalsteps. The analyzer obtains the advantages of both the digital computersand the analog diierential analyzers. The analyzer has the advantages ofa digital computer in that it produces a quick and accurate solution ofmathematical problems. The apparatus also includes the advantages of ananalog differential analyzer in that it requires a minimum number ofcomponents to solve differential equations. because of the logicalsystem of component operation which has been incorporated into themachine. Because of these advantages, the analyzer requires only arelatively small space to obtain the solution of complex dif# ferentialequations.

This invention provides apparatus which is included in the digitaldilferential analyzer to enhance the accuracy of the solutions obtainedby the analyzer. The apparatus operates with the analyzer to reduce by afactor of 2 the errors produced by the analyzer in the solution of manyproblems, and fairly often the apparatus operates to reduce the error bya factor substantially greater than 2. The apparatus is relativelysimple and requires only a relatively small increase in the spaceoccupied by the analyzer. The apparatus of the present inventionprovides a Variation in the organization of a digital differentialanalyzer to enable either positive, negative or zero values asrepresented by signal indications. Because of this ternary method ofindication, a digital differential analyzer is able to provide adetermination on a ternary basis of the values of numerical quantitiesAn object of this invention is to provide a system for operating inconjunction lwith a digital diiferential analyzer to minimize the errorsproduced by the analyzer in solving a problem.

Another object is to provide apparatus of the above character which canbe easily incorporated in a digital differential analyzer to enhance theaccuracies in the solutions obtained by the analyzer.-

A further object is to provide apparatus of the above character whichrequires a minimum number of components so that only a relatively smallincrease is required in the size of the analyzer to obtain the benelitsfrom the apparatus.

Other objects and advantages will be apparent from a detaileddescription of the invention and from the appended drawings and claims.

In the drawings:

Figures 1, 2 and 3 are schematic diagrams, partly in block form andpartly in perspective, showing the features which together constitute adigital differential analyzer and which operate in conjunction with thedigi- The number of components are further reducedl 2,900,135 PatentedAug. 18, 1959 ICC tal differential analyzer to forrn one embodiment ofthis invention;

Figure `4 is a bloclg diagram illustrating the ,operation of one of theintegrators forming a part of the' digital diierential analyzer shown inthe previous figures;

Figure 5 is a curve illustrating the operation f the integrator shown inFigure 4; I y

Figure 6 is a chart which illustrates how d ilferent parts of theintegrator such as that shown in Figure 4 are coded to control theoperation of the integrator; l

Figure 7 is a schematic diagram illustrating the relationshipbetweendifferent integrators forming the digital dif-l ferential analyzer shownin Figures 1 to 3, inclusive, when the analyzer is solving a particularproblem; and

Figure 8 is a chart illustrating the operation of certain Aof thecomponents shown in Figures 1 to 3, inclusive.

The digital differential analyzer shown in Figures l to coating 12 ofmagnetic material is applied to the periphery of the drum. The coating12 can be considered as being divided into a plurality of annularchannels 14, 16, 18, 20 and 22. Each of the channels is separated by asufficient distance from its adjacent channels so as to be substantiallyunaffected by the magnetic information provided in the adjacentchannels.

The circumferential distance of each channel may be considered as beingdivided into a plurality of positions. Each of the positions issufficiently separated from its adjacent positions to receive adifferent magnetization than that provided on the adjacent positions.For eX- ample, approximately 1160 equally spaced pulse positions may beprovided in each channel when the drum has a radius of approximatelyfour inches.

A plurality of toroidal coils are positioned adjacent to each of thechannels 14, 16, 18, 20 and 22. For eX- arnple, coils 24, 26 and 28 areprovided in contiguous relationship to the channel 14. Similarly, coils30, 32 and 34; coils 36, 38 and 40; and coils 42, 44 and 46 areassociated with the channels 16, 18 and 20, respectively. A single coil50 is disposed adjacent the channel 22. Amplifiers (not shown) may beassociated with each of the coils in the different channels.

The coils 24 and 28 are effectively separated from each other byapproximately 184 pulse positions, and the coil 26 is disposed at anintermediate position between the coils 24 and 28. The coil 28 isadapted to provide signals in a pattern dependent upon the operation ofthe digital differential analyzer and to produce a correspondingmagnetic pattern on the drum 10 as the drum rotates. The patternproduced on the drum 10 by the coil 28 is of the binary form in which amagnetization in one circumferential direction indicates one value and amagnetization in the other direction indicates a second value.

The coil 24 is adapted to pick up the changes in the direction ofmagnetization in the channel 14 as the drum rotates. The coil 26 isadapted to produce a substantially constant signal for returning thedirection of magnetization on the drum to that representing a 0 valueafter the magnetic pattern on the drum has been converted into acorresponding electrical pattern by the coil 24.

The coils 30, 32 and 34 are separated from one another by distancescorresponding to the distances between the coils 24, 26 and 28 and areadapted to perform functions similar to those performed by the coils 24,26 and 28, respectively. The coils 38 and 44 are also adapted to operatein a manner similar to the coil 28 to provide a magnetic pattern in thechannels 18 and 20, respectively, in a pattern dependent upon theproblem to be solved.

The coils 38 and 44 are effectively separated from the coils 40 and 46,respectively, by approximately 49 pulse positions during the operationof the analyzer to obtain thesolution of a mathematical problem. Thecoils40 and '46 are adapted to produce signals in accordance.

with the magnetic pattern provided in their respective channels by thecoils 38 and 44. The coils 36 and 42 are adapted to operate in a mannersimilar to the coil 26 ,to produce a constant zero direction ofmagnetizae tion in the channels 18 and 20, respectively, after thepattern provided by the coils 38 and 44 have been utili'zed'by the coils40 and 46, respectively.

The coil 50 is adapted to produce a cycle of a signal approximating asine wave as each pulse position in the channel 22 moves past the coil.The coil 50 produces. a pattern-of sine waves because of the magneticpattern permanently provided in the channel 22. This pattern remainsconstant regardless of the problem to be solved.

A counter 52 is connected to the coil 50 to count the cycles of sinewaves in the channel 22 as the drum 10 rotates. The counter 52 is formedfrom a plurality of multivibrators connected in cascade arrangement andis adapted to count successive sine waves in a numerical range from l to48. Upon each count of "48, the counter 52 is adapted to return to itsnew state for the initiation of a new count. As will be disclosed indetail hereinafter, a new integrator is presented for computation uponthe completion of each count of 48 by the counter 52. The counter V52may be constructed in a manner similar to that shown in Figures 17 and18 ofthe drawings and disclosed on pages 75 to 78, inclusive, of thespecification for copending application Serial No. 217,478 filed March26, 1951, by Floyd G. Steele and William F. Collison.

Similarly, a counter 54 is formed from a plurality of multivibrators incascade arrangement. The counter 54 is connected to the counter 52 tocount the number of times that a full count is vobtained in the counter52. For example, the. counter 54 may count up to 22 Vfull counts in thecounter 52 before returning to its initial state for the initiation of anew count. In this way, the` counter 54 divides the drum 10 into 22integrators each having 48 pulse positions. The counter 54 may beconstructed in a manner similar to that shown in Figures 24 and 25 ofthe drawings and disclosed on pages 83V to 85,`inclusive, of thespecification for copending application Serial No. 217,478.

The coils 24, 30, 40 and 46 are connected to the grids of the left tubesin bistable multivibrators 56 (Figure l), 58,A 60 (Figure 3) and 62,respectively. The4 bistable multivibrators 56', 58 and 60 areconventional circuits and may be constructed in a manner similar to thatdisclosed on page l of the specilication for copending applicationSerial No. 217,478. Connections are also made from .the coils 24, 30, 40and 46 to the input terminals of inverters 66, 68, 70 and 72, the outputterminals of which are connected to the grids of the right tubes in themultivibrators 56 (Figure l), 58, 60 (Figure 3) and 62, respectively.The inverters 66, 68, 70 and 72 may be constructed in a manner similarto the 'rst stage shown in Figure 9 of the drawings and disclosed onpage 49 of the specication for copending application Serial No. 217,478;

Gate circuits 76 (Figure 2), 78 (Figure 3), 80 (Figure 2) and 82 areconnected to the plates of the left tubes in the multivibrators 56, 58,60 and 62, respectively. As will ebe disclosed in detail hereinafter,each of the gate circuits 76, 78, 80 and 82 operates to pass outputsignals only when all of its input terminals simultaneously receive`lrelatively high voltages. minology, gate circuits similar to thecircuits 76, 78, 80 and 82 are known as and circuits or networks. Forpurposes of convenience, such circuits are shown as rectangles in theattached drawings. The gate circuits 76,

78, 80 and 82 may be constructed in a manner similar In the computerter-g to that shown in Figure 11 of the drawings and disclosed on pages56 and 57 of the specification for copending application Serial No.217,478.

The gate circuits 76 (Figure 2) and 78 (Figure 3) are connected througha line 83 to an output terminal of the counter 52 so as to become openfor the passage of information from the coils 24 and 30, respectively,only yduring the rst 22 pulse positions of each integrator.

' The output signals from the gate circuits 76 and 78 are introduced toinput terminals of networks 84 (Figure 2) and 86 (Figure 3), the outputterminals of which are connected to the coils 28 and 34, respectively.Eacln of the networks 84 and 86 is adapted to pass output 'signals whenrelatively high voltages are introduced to any one of the inputterminals in the network. In the computer terminology, networks similarto the networks- 84 and 86 are designated as or networks. For purposesof convenience, such networks are illustrated as triangles in,theatta'ched drawings. The or networks 84 'and 86 may be constructed ina manner similar to that shown in Figure V10 of the drawings anddisclosed on pages 55 and 56 of the specification for copendinglapplication Serial No. 217,478'.

including several channels, used to provide the storage Y for anintegrator. The output signals from the gate circuits 80 an'd 82 areintroduced to or networks 90 and 92having their output terminalsconnected to the coils i 38'and 44, respectively.

A gate circuit 94 (Figure l) has input terminals connected to the platesof the left tubes in the multivibrators 56'and '62. The gate circuit 94also has an input terniinal connected through the line 83 rto anappropriate output terminal of 'the counter 52. The output from the gatecircuit 94 is introduced to the grid of the left tube in a bistablemultivibrator 96. The grid of the right tube in the multivibrator 96 isconnected through a line 97 to an output terminal of the counter 52 suchthat the `tube becomes cut oi at the 48th pulse position of eachintegrator. The line 97 is also connected to an input terminal of thegate circuit 76 (Figure 2).

The plate of the left tube in the multivibrator 96 is connected to gatecircuits 98 (Figure 1) and 100. A second input terminal of the gatecircuit 98 also has voltage applied to it from the output terminal of agate circuit 102, and a second input terminal of the gate lcircuity 100has voltage applied to it from the output terminalof a gate circuit104'.Y Y

Input terminals of the gate circuit 102 are connected to the plate ofthe left tube in the multivibrator 58 and through a line 106 to anoutput terminal of the counter 52, so as to :become opened after the22nd pulse position for each integrator. Connections are made to inputterminals of the gate circuit 104 from the plate of the right tube inthe multivibrators 58 and from the line 106.

Similarly, the voltage on the plate of the light tube in themultivibrator 96 is introduced to input terminals of gate circuits 108and 110. The gate circuits 108 and 110 also have input terminals whichare connected to the output terminals of the gate circuits 102 and 104,respectively. Connections are made from the 'output terminals of thegate circuits 108 and 11`0 to input terminalsof or networks 112 and 114,respectively. Volt` 4agesare'also introduced to the'networks'112 and 114from theoutput fterrninals of -ga'te circuits v100 and 98, re-rspectively.

`he i'tput signals from the or networks 112 and.114

are introduced to input terminals of gate circuits 116y and 118, otherinput terminals of which are connected to the plate of the left tube ina bistable multivibrator 120. The operation of the left tube in themultivibrator 120 is controlled by a voltage introduced to its grid fromthe output terminal of the gate circuit 102. The voltage introduced tothe grid of the right tube in the 'multivibrator 120 through the line 97from the counter 52 controls the operation of the right multivibratortube.

Connections are made from the output terminals of gate circuits 116 and118 to input terminals of an adder 126. The adder 126 may be constructedin a manner similar to that shown in Figure 45 of the drawings `arnddisclosed on pages 118 to 120, inculsive, of the specification forcopending application Serial No. 217,478. Other input terminals of theadder 126 are connected to the output terminals of a carry circuit 128and of gate circuits 130 and 132. The carry circuit 128 may beconstructed in a manner similar to that shown in Figure 46 of thedrawings and disclosed on pages 118 to 122, inclusive, of theyspecification for copending application Serial No. 217,478. The gatecircuit 130 has signals introduced to it from the plate of the left tubein the multivibrator 56 and from the line 106 connected to the counter52. Similarly, signals are introduced to the gate circuit 132 from theline 106 and from the plate of the right tube in the multivibrator 56.

The output signals from the adder 126 are introduced to an inputterminal of a gate circuit 138 (Figure 2), which also has inputterminals connected to the plate of the left tube in a multivibrator 140and through the line 106 to the counter 52. The grid of the left tube inthe multivibrator 140 is connected to the output terminal of a gatecircuit 141, input terminals of which are connected to the plates of theleft tubes in the multivibrators 56 and 60 and through the line 83 tothe counter 52.

The output terminal of the gate circuit 138 is connetced to an inputterminal of an or network 142 having it output terminal connected to theor network. 84. A second input terminal of the or network 142' receivessignals from a gate circuit 143, input terminals of which are connectedto the plate of the left tube in the multivibrator 56, to the plate ofthe right tube in the multivibrator 140 and through the line 106 to thecounter 52. The grid of the right tube in the multivibrator 140 isconnected through the line 97 to an output terminal of the counter 52.

In addition to being applied to the adder 126, signals are also appliedfrom the carry circuit 128 to a gate circuit 146. Connections are madeto other input terminals of the gate circuit 146 from the plate of theleft tube in the multivibrator 140, the output terminal of the ornetwork 112 and through the line 97 from the position counter 52.

' The output from the carry circuit 128 is also introduced to an inputterminal of a gate circuit 152. Other input terminals of the gatecircuit 152 are connected to the plate of the left tube in themultivibrator 140, the line 97, and the output terminal of the ornetwork 114.

Connections are made from .the output terminals of the gate circuits 146and 152 to the input terminal of and or network 154, the output terminalof which is connected to the or network 90. The output terminals of thegate circuits 146 and 152 are also connected to input terminals of gatecircuits 158 and 160, respectively, other input terminals of the gatecircuits 158 and 160 being connected to the plates of the right and lefttubes in the multivibrator 56, respectively. The output signals from thegate circuits 158 and 160 are connected to the input terminals of an ornetwork 162, the output from which is applied to an input terminal ofthe or network 92.

`.In addition to its previously disclosed connections,v

the plate of the left tube in the multivibrator 58 is con-'- nected to.an input terminal of a gate circuit 166 (Fig-- ure 3). Connections aremade to other input terminals of the gate circuit 166 from the plate vofthe left tube in the multivibrator 60 and through the line 83 from theposition counter V52. The output from the gate circuit 166 is applied toinput terminals of gate circuits 168 and 170, other input terminals ofwhich are connected to the plates of the left and right tubes in themultivibrator 62, respectively. y

Connections are made from the output terminals of the gate circuits 168and 170 to input terminals of a counter 172. The counter 172 is formedfrom a plurality'of multivibrators connected in cascade arrangement toprovide a resultant indication of a plurality of increments. The outputterminals of the counter 172 are connected to the input terminals of astepping circuit 174, the operation of which is initiated upon theintroduction of a high voltage from the plate of the left tube in themultivibrator 120. The counter 172 and the stepping circuit 174 may beconstructed in a manner similar to that shown in Figure 29 of thedrawings and disclosed on pages to 93, inclusive, of the specication forcopending application Serial No. 217, 478.

The output signals from the stepping circuit 174 are introduced to anadder 176. The adder 176 may be constructed in a manner similar to thatshown in Figure 38 of the drawings and disclosed on pages 108 and 109 ofthe specification for copending application Serial No. 217,478. Othersignals are introduced to the` adder 176 from the gate circuits 102 and104, from a carry circuit 178 and from the plate of the left tube in themultivibra! tor 120. The output from the adder 176 is applied to inputterminals of the carry circuit 178 and the or net' work 86. The carrycircuit 178 may be constructed in a` manner similar to that shown inFigure 39 of the drawings and pages 108 and 109 of the specification.for copending application Serial No. 217,478.

The digital differential analyzer disclosed above is adapted to providethe solution of differential equations. For example, it may provide thesolution for a general equation y=f(x) so as to obtain a function fydx:f f (x) dx, where f(x) represents a function of x and ff(x)dx representsthe integral of the function. If a curve y=f(x) is plotted with x as theabscissa and y as the ordinate, the analyzer obtains the relationshipfydx=ff(x)dx by computing the area under the curve y=f(x). Bydetermining the area under the curve y=f(x), the analyzer performselectronically operations that may sometimes be performed mentally by askilled mathematician when the problem to be solved is relativelysimple.

The analyzer obtains the value of lthe function fydx=ff(x)dx byproducing small increments of x. These increments may be represented bythe symbol Ax. For each Ax increment, the analyzer determines the valueof y and obtains the product yAx. This product yAx i.e. the differentialcombination of y and Ax, represents the area under the curve y=f(x) foreach Ax increment, as indicated in Figure 5 by the shaded area 200 for aparticular Ax increment. If the product yAx is obtained for successiveAx increments and if all of the yAx increments are added together, thearea under the integral of the curve representing f (x) from x0 to x maybe approximated. The approximation may be as close to the actual valueas desired by decreasing the value of each Ax increment.

An integrator for determining the yAx increments and for storing thecumulative values of these increments is shown in Figure 4. Theintegrator includes a transfer stage 202 for obtaining Ax increments atperiodic intervals through a line 204. The transfer stage 202 isequivalent in one embodiment to the channel 14, the coils 24, 26 and 28associated with the channel, the gate circuit 141, the bistablemultivibrator and other members and stages, as will become apparenthereinafter bydetailed disclosure. The integrator also. has an integrandaccumulator 206 for 4storing the valueof the dependent quantity y andfor receiving Ay increments through a lineY 208. from its own and fromother integrators so ask to vary the value of y in accordance with thefunction y=f(x). The integrand accumulator 206 is equivalent in oneembodiment to the channel 16, the coils 30, 32 and 34 associated withthe channel, the gate circuits 166, 168 and 170, the counter 172 andstepping circuit 174, the adder 176, the carry circuit 178 and othermembersand stages, as will become apparent hereinafter by detaileddisclosure. An output accumulator 210 is provided to receive yAxincrements, to combine each yAx increment with the previous incrementsand to deliver the cumulative value obtained to another integrandaccumulator or transfer stage while holding the remainder'in store. Theoutput accumulator is equivalent in one embodiment to the channel 14,the coils 24, 26 and 28, the gate circuits 98, 100, 108,- 110, 116 and118, the adder 126, the carry circuit 128 and other members and stages,as will become apparent hereinafter by detailed disclosure.

The interrelationship between different integrators is illustrated inFigure 7 for a particular problem represented by As is mathmeticallyknown, the diierential solution of this problem indicates that y=tan x.The integrators involved in the solution of this problem are indicatedin Figure 7 by blocks 212, 214 and 216. The integrators 218 and 220 arethen utilized to obtain the function x tan x from the function tan xgenerated by the integrators 212, 214 and 216. In each integrator, theintroduction of the Ax increments constituting the independent variablefor the integrator is indicated by a line extending into thel block atthe upper right side of the block. The Ay increments are introduced intothe integrator through a line or a plurality of lines extending into thelower right portion of the block representing the integrator. The outputof the integrator is obtained from-a line extending from an intermediateposition at the right side of the appropriate block.

As will be seen in Figure 7, Ax increments of the independent variablefor a particular integrator may be obtained from the output of anotherintegrator. For example, in Figure 7, the Ax increments for theintegrators 214 and 216 are obtained from the output of the integrator212. Similarly, Ay increments for a particular integrator may beobtained from the output of other integrators as well as from the outputof the integrator itself. For example, Ay increments ,for theintegrators 214 and 218 are obtained from the output of the integrator212.

The Ax and Ay increments for each integrator are actually determinedfrom a coded pattern provided in an integrator storage section of thechannels 14 and 16, respectively. As previously disclosed, the pulsepositions in each channel are subdivided into `22 integrator vstoragesections each having 48 pulse positions. The rst 22 positions in eachintegrator storage section in the channel 14 are coded to indicate a Axincrement. Since the first 22 positions in the channel 14 for eachintegrator storage section correspond in number to the 22 integrators inthe analyzer, the pulse representing a Ax increment from each integratorstorage section is recorded in a particular position in the channel 14.This position corresponds to the particular integrator from which the Axincrements are obtained. For example, the Ax increments for theintegrator 214 in Figure 7 would be coded in a particular o ne of the 22positions in the channel 14 corresponding to the time at which theoutput from the 'integrator 212 appears on the coils 40 and 46. InFigure 6, a pulse 222 is shown as being recorded in the channel 14 inthe llth pulse position for a particular integrator.

A pulse in the channel 14 in one of the iirst 22 positions for aparticular integrator indicates that a Ax increment may b'e made for theintegrator. However, such a presenceor absence of a coincidental pulsein the channel 18. If a positive pulse is picked up from the channel 18by the coil 40 atthe same time as the pulse representing a possible Axincrement for a particular integrator is picked up by the coil 24, a Axincrement for the in-v tegrator actually occurs. For example, the pulse222' in Figure 6 indicates an actual Ax increment for a particularintegrator sinceV itcoincides in time 4with a pulse 224 in the channel18. A Ax increment is not obtained for the integrator if a pulse doesnot appear in the channel 18 at the same time as the pulse in thechannel 14.

The polarity of each Ax increment is determined by the presence orabsence ofv a coincidental pulse in the chan-v nel 20. If a pulse ispicked. up from the channel 20 by the coil 46 at the same time thatpulses indicating an' actual Axincrement for a particular integrator arepicked up by the coils 24 and 40, .the Ax increment for the integratorVis positive. The Ax increment is negative if a pulse does not appear-iu the channel 20 at the same time as the pulses in the channels 14 and18. For example, the pulse 222 in-Figure 6 indicates a negative Axincrement since a pulse does not appear in the channel 20 simultaneouslywith the occurrence of the pulses 222 and 224 in the channels 14 and 18,respectively.

The first 22 positions in the channel 16 for each integrator storagesection are coded to indicate Ay increments in tions in the channel 14to indicate Ax increments. Since.

the rst 22 positions in each integrator storage section correspond tothe 22 integrators in the digital differential analyzer, each integratoris coded in particular ones of the first 22 positions in the channel 16so as to receive the outputs from certain other integrators inaccordance with the problem to be solved. For example, a pulse would becoded in the channel 16 in a particular one of the tirst 22 positionsfor theintegrator 216 in Figure 7 so as to coincide with the time atwhich the output from the integrator 212 is made available to the coils40 and 46 in the channels 18 and 20, respectively. Although only one Axincrement can be obtained from an integrator storage section upon eachrevolution of the drum, several Ay increments can be obtained. This maybe seen by the pulses 232 and 234 in the channel 16 in Figure 6.

Each pulse in the first 22 positions in the channel 16 for eachintegrator represents the possibility of a Ay increment but does notindicate the actual occurrence of such an increment or the polarity ofthe increment. The actual occurrence of the increment is indicated bythe presence or absence of a pulse in the channel 18 at the same timethat the pulse in the channel 16 is made available to the coil 30. Forexample, the pulse 232 in Figure 6 indicates an actual Ay `increment fora particular integrator sinceit coincides in time with a pulse 236 inthe channel 18. However, no Ay increment is obtained when the pulse 234is picked up by the coil 30 since there is no coincidental pulse in thechannel 18.

The sign of each actual Ay increment is indicated by the presence orabsence of a pulse in the channel 20 at the time that pulses in thechannels 16 and 18 are simultaneously made available to the coils 30 and40. For example, the pulse 232 in Figure 6 indicates a positive Ayincrement for a particular integrator since a pulse 238 appears in thechannel 20 at the time that the pulses 232l and 236 are picked up by thecoils 30 and 40, respectively.

Since the interrelationship between the different integr-ators remainsconstant during the solution of a particular problem, the codingpulsesin the channels 14 and 16 for the first 22 positions of each integratormust be re- 4 tained during the computation. Retention of the pulses inthe channel 14 is provided by the multivibrator 56 (Figure 2), the gatecircuit 76 and the or network 8f4. The pulses in the channely 14 havinga particular direction of magnetization are converted by the coil 24 topulses of relatively high voltage. These voltage pulses are thenintroduced to the grid of the left tube in the multivibrator 56 so as tocut off the tube. When the left tube in the multivibrator 56 becomes cutolf, a high voltage is produced on the plate of the tube and isintroduced to the gate circuit 76.

The gate circuit 76 is opened by a signal from the counter 52 rwhen thefirst pulse from each integrator storage section is picked up by thecoil 50. The gate circuit 76 remains open so that information in thechannel 14 up to and including the 22nd pulse position of eachintegrator storage section can pass through the gate circuit. During thetime that the gate circuit 76 remains open, the positive pulses from theplate of the left tube in the multivibrator 56 pass through the gatecircuit to the or network 84. The network 84 in turn passes to the coil28 any positive pulses introduced to it, and the coil 28 operates toproduce a magnetic pattern in the channel 14 similar to that introducedto the network 84.

Similarly, the multivibrator 58 (Figure 3), the gate circuit 78 and theor network 86 operate to recirculate in the channel 16 the informationprovided in the rst 22 positions of each integrator. As previouslydisclosed, pulses lmay be provided in the channel 16 in the first 22pulse positions for each integrator to indicate whether any variationsin the value of the dependent quantity y will be made for theintegrator.

Since the gate circuit 78 remains open during the first 22 positions foreach integrator and since all of the Ay increments for the integratoroccur in the rst 22 positions, all of the coding pulses controlling theproduction of the Ay increments are retained for subsequent utilization.These coding pulses correspond to the pulses 232 and 234 in Figure 6 andcontrol by their positioning the interrelationship between the dependentquantity for the integrator undergoing computation and the outputquantities from certain of the integrators. This interrelationship isdependent upon the problem requiring solution, as may be seen in Figure7 for a particular problem. In this way, each integrator can receive Ayincrements from a plurality of interrelated integratorsin accordancewith a problem to be solved.

The gate circuit 166 (Figure 3) operates to determine whether or not anactual Ay increment is made for an integrator at the time that a codingpulse appears in the channel 16 in one of the rst 22 pulse positions ofan integrator storage section. The gate circuit 166 receives the codingpulses in the channel 16 because of its connection to the plate of theleft tube in the multivibrator S8. The connection from the counter 52through the line 83 to the gate circuit 166 causes the gate circuit tobecome operi only during the first 22 positions of each integratorstorage section. Since the gate circuit 166 is also connected to theplate of the left tube in the multivibrator 60, it can open for thepassage of a signal only when high voltages are simultaneously producedon the plates of the left tubes in the multivibrators 58 and 60.

A relatively high voltage is produced on the plate of the left tube inthe multivibrator 60 only when a relatively high voltage is induced inthe coil 40. As will be disclosed in detail hereinafter, the coil 40indicates in adjacent pulse positions the cumulative value of thesuccessive yAx increments for each of the 22 integrators in theanalyzer. The cumulative values of the yAx increments for the dierentintegrators are made available by the coil 40 to each integrator toprovide an indication of the Ay increments for the integrator upon thepresentation of the coinciding integrator storage section forcomputation. For example, Ay increments for the integrators 214 and 218in Figure 7 are obtained in accordance with the cumulative value in thechannel 18 of the yAx increments of the integrator 212. The productionof -a pulse by the coil 40 at the same time as the appearance of acoding pulse in the channel 16 indicates only the occurrence of a Ayincrement, but it does not indicate whether such lincrement is positiveor negative.

The pulses produced by the -gate circuit 166 to indicate Ay incrementsfor the diii'erent integrators are introduced to the gate circuits 168and 170. The gate circuit 168 also has` a vol-tage introduced yto itfrom the plate of the ileft tube in the multivibrator 62. This voltageis high when the left tube in the multivibrator 62 becomes cut off uponthe introduction of positive pulses from the coil `46 tothe grid of thetube. Positive pulses are induced in the coil 46 at the same time thatthe positive pulses in the channel 18 are induced in the coil 40. Thesimultaneous occurrence of pulses in the channels 18 and 20 indicatesthat the pulse in the channel 18 has a positive value.

When positive pulses are simultaneously introduced to the gate circuit168 Ifrom the gate circuit 166 and the plate of the left tube in themultivibrator 62, the gate circuit 168 passes a signal to the counter172.. This signal causes the numerical indications provided by thecounter 172 to increase by an integer ina positive direction. Forexample, a signal passing to the counter 172 from the gate circuit 168may cause the counter to provide a numerical indication of +4 when anindication of +3 was previously provided by the counter. Similarly, theindications in the counter 17.2 may change from -4 to +3 upon theintroduction of a signal from the gate circuit 168.

As previously disclosed, a positive increment in the cumulative yAxvalue for an integrator is indicated by the simultaneous occurrence ofpulses of the same polarity in the channels 18 and 20. Similarly, anegative increment in the cumulative yAx value for the integrator isindicated by the simultaneous production of pulses of opposite polarityin the channels 18 and 20. Because of the particular polarity of thepulse in the channel 20, a relatively low voltage is induced in the coil46. This voltage is inverted by the inverter 72 and is introduced as arelatively high voltage to the grid of the right tube in themultivibrator 62. This voltage causes the right tube in themultivibrator 62 [to become cut oi and a relatively high voltage to beproduced on the plate of the tube.

When a relatively high voltage is produced on the plate of the righttube in the multivibrator 62 at the same time that a positive pulsepasses through the gate circuit 166, the gate circuit 170 opens andpasses a signal to the counter 172. This signal provides an indictationof a negative Ay increment. Since the counter 172 is adapted to providea negative count as Well as a.y positive count of the Ay increments, itoperates upon the introduction of a signal from the gate circuit 170 tosubtract an integer from fthe resultant value in the counter. l Forexample, the indications in the counter 172 are changed from +4 to +3when a signal is introduced to the counter from the gate circuit 170.Similarly, the counter 172 may provide an indication of -5 upon theintroduction of a signal from the gate circuit 170 .at a time when thecounter has previously provided an indication of 4 4. 'Ilhe operation ofa counter similar to the counter 172 in providing a positive andnegative count of digital increments is fully disclosed in co-pendingapplication Serial No. 217,478 filed March 26, 1951,

. by Floyd G. Steele and William F. Collison.

H55 at 'the right.A In binary' form, a pattern* of 101 india valueA of+3 is indicated by a pattern of 01'1, where the'least signicant digit isat the right.

The resultant value of Ay increments accumulated in the counter 172 foreach integrator isk made available on a step by step basis by thestepping circuit 174 which feeds the information sequentially intol theadder 176.V

For example, when the resultant value of the Ay increments for aparticular integrator is +5, the circuit 174 indicates a value of +1upon the rotation of the drum 10 past the pulse position which indicatesthe least signicant digit of the number. This corresponds to the valueof the least signficant digit in the binary'V indication of +5. As thedrum rotates past the second and third pulse positions, the circuit 174indicates values of O and 1, respectively. Similarly, for a Ay incrementof +3 for a particular integrator, the circuit 174 indicates successivevalues of 1, 1 and 0 as the drum 10 rotates through the pulse positionsindicating the three least significant digits of the value of y for theintegrator. The operation of a stepping circuit similar to the circuit174 and of an adder similar to the adder 176 is fully disclosed inco-pending application Serial No. 217,478.

The stepping circuit 174 operates to pass in sequence the binaryindications in the counter 172 only dur-ing the time that a relativelyhigh voltage is produced on the plate of the left tube in themultivibrator 120. The left tubein the multivbrator 120 is cut off uponthe introduction of a signal from the gate circuit 102. The gate crcuitis so connected to lthe counter 52 that it cannot open -for the passageof a triggering signal until after the 22nd pulse position in thechannel 16 for each integrator. When the first pulse appearsin thechannel 16 for an integrator after the initial 22 positions for theintegrator, the gate circuit 102 is opened for the passage of atriggering signal to cut oif the left tube in the multivibrator 120.

The production of a relatively high vol-tage on the plate of the lefttube in the multivibrator 120 causes the adder 176 as wel-l as thestepping circuit 174 to be triggered into operation. After the adder 176has been triggered into operation, it receives binary indications of thevalue of the dependent quantity y for each integrator and arithmeticallycombines these indications with the values of Ay passing through thecircuit 174. The adder 176 receives indications representing the integer1 from the gate circuit 102 and indications representing the value Hfrom the gate circuit 104. The operation of the vgate circuits 102 and104 is in turn controlled by the voltages on the plates of the left andright tubes in the multivibrator 58.

The arithrnetical combination of the values of y and Ay are obtained foreach pulse position in sequence as the drum i rotates. For example, Ithearithmetical combination of the indications of y and Ay in the 25thpulses position for a particular integrator may first be obtained; Thearithmetical combination of the values of y and Ay may thereafter besequentially obtained for the 26th, 27th and the following pulsepositions for the integrator, eg. the pulse positions in the channelscom.

stituting an integrator storage section.

Sometimes upon the arithemtical combination of the values of y and Ayfor a particular pulse position, the adder 176 may obtain a Afull'binary indication of +2. In binary form, an indication of +2 isequivalent to a value of zero for the pulse position and a carry of `+1to the next highest digit. For example, when a binary indicationV ofl-l-l for y in the 26th position is added toa binary indication of +1for Ay in the same position, the resultant value may be zero in the 26thpositionl with a carry of +1 into the 27th position` This carry isAprovided by the circuit 1,78.

A carry may also be provided from a trstpulsel position to the nextposition when a vcarry from the position"v immediately preceding thefirst position is added to the integer l indicating the value of eithery or Ay for the rst position. For example, a carry may be provided frompulse position 29 to pulse position 30 as a result of an addition invpulse position 29. The addition of this carry indication with an integerl indicating the value of the dependent quantity y for pulse position 30causes a carry to be obtained for pulse position 31.

By arithmetically combining the values of y from the multivibrator 58,the values of Ay from the stepping circuit 174 and the carry indicationsfrom the circuit 178 for each pulse position, an new value of y isobtained for each pulse position. The new indication of y for each pulseposition passes sequentially through the or network 86 and produces acorresponding signal pattern in the coil 34. This signal pattern causesthe coil 34 to record in the channel 16 the new value of y for eachpulse position. The information relating to the new value of y issubsequently utilized by the adder 176 as it moves past the coil 30 andit is thereafter erased by the coil 32.

The occurrence of each Ax increment and the polarity 0f each suchincrement are determined in a manner similar to that disclosed above forthe Ay increments. Thus, a Ax increment for an integrator occurs when apulse of relatively high voltage is induced in the coil 40 at theinstant that a pulse is induced in the coil 24 in one of the first 22positions for the integrator. Similarly, the Ax incrementV is positivewhen a pulse is induced -in the coil 46 at the instant that the'codingpulse for the integrator is induced in the coil 24. The Ax increment isnegative if a pulse is not induced in the coil 46 at the instant thatthe coding pulse for the integrator is induced in the coil As previouslydisclosed, the pulses induced in the coil 24 are introduced to the gridof the left tube in the multi-vibrator 56 (Figures 1 and 2) to cut offthe tube in a pattern corresponding ot the pattern of magneticl pulsesin the channel 14. The positive pulses produced on the plate of the lefttube in the multivibrator 56 are introduced to the gate circuit 94.Since the gate circuit 94 is also connected through the line 83 to thecounter 52,. it is prepared to open for the passage of a signal duringthe first 22 pulse positions of each integrator. Because of itsconnection to the plate of the left tube in the. multivibrator 62, thegate circuit becomes open only when a pulse appears in the channel 20simultaneously with the appearance of the coding pulse in the channel14. Such a simultaneous occurrence of pulses in the channels 14 and 20for an integrator indicates that a Ax increment for the integrator ispositive.

Upon the passage of a signal through the gate circuit 94, the left tubein the multivibrator 96 (Figure l) becomes cut oif. The left tube in themultivibrator 96 remains cut off during the rest of the time that theparticular integrator is presented for computation. At pulse position 48of the integrator, a signal is introduced from the counter 52 throughthe line 97 to the grid of the right tube in the multivibrator 96 so asto cut oft" the tube; When the right tube in the multivibrator becomescut oft, the left tube starts to conduct. This causes the left tube inthe multivibrator 96 to be prepa-red for triggering by a signal passingthrough the gate circuit 94 during the tirst 22 positions of' the nextintegrator to be presented for computation.

When the left tube in the multivibrator 96 tis cut on, a relatively highvoltage is introduced from the plate of the tube to input terminals ofthe gate circuits 98 (Figure 1) and 100. The gate circuit 98 alsoreceives signals from the gate circuit 102, which operates after thefirst 22A positions for each integrator to pass the pulses of relativelyhigh voltage induced in the coil 30. As previously disclosed, thesepulses provide an indication representing.

the integer .lf for different pulse positions each inte- L 18 grato?.Since each indication represents the dependent. qauntity y for eachintegrator, they may b e designated as Y in conformity with thedesignations provided in copending application Serial No. 217,478.

When relatively high voltages are simultaneously introduced to the gatecircuit 98 from the gate circuit 102 and the plate of the left tube inthe multivibrator 96, the gate circuit passes a signal to the or network114. Since the gate circuit 102 passes signals indicative of Y and theleft tube in the multivibrator 96 indicatesv +Ax when its Voltage ishigh, each signal passing to the or network 114 indicates the value (Y)(Ax) for a particular pulse position. The value (Y) (Ax) is an andproposition which is true only when both Y and Ax are simultaneouslytrue. The value (Y) (Ax) corresponds in binary form to the integer l forthe pulse position.

For a value of for a pulse position, a relatively low voltage is inducedin the coil 30 at the pulse position. This loW voltage is inverted bythe inverter 68 and is introduced as a relatively high voltage to thegrid of the right tube in the multivibrator 58 to cut oi the tube. Theresultant positive pulse on the plate of the right tube in themultivibrator 58 passes through the gate circuit 104. In this Way, thepulses passing through the gate circuit 104 indicate the value of Y fordifferent pulse positions, where Y' is the inverse of the value of Y andindicates the integer 0 for the different pulse positions. This causesthe signals passing through the gate circuit 100 to provide anindication of (Y) (Ax) for the diierent pulse positions. These signalspass to the orl network 112.

Since the left tube in the multivibrator 96 is triggered into a state ofnon-conductivity only for positive Az increments, the right tube in themultivibrator remalns cut ofi for negative Ax increments. When the Axincrements are negative, a relatively high voltage is introduced fromthe plate of the right tube in the multivibrator 96 to the gate circuits108 and 110. Because of its connection to the gate circuit 102, the gatecircuit 108 passes signals only for those pulse positions when arelatively high voltage is induced in the coil 30. Thus the signalspassing through the gate circuit 108 provided an indication of (Y)(-Ax).As is well known, (Y)(-Ax) is arithmetically equivalent to (-Y) (Ax). Aswill be disclosed in detail hereinafter, -Y is equivalent to +Y. Thusthe signals passing through the gate circuit 108 effectively provide anindication of (Y) (Ax) and correspond to the signals passing through thegate circuit 100. The signals from the gate circuits 100 and 108 may bedesignated as Ya in a manner similar to that disclosed in co-pendingapplication Serial No. 217,478 and shown in Figures 45 and 46 of theco-pending application. These signals are introduced to the gate circuit116 after passing through the or network 112.

In like manner, the signals passing through the gate circuit 110 providean indication of (Y)(-Ax). Since Y is equivalent to -Y Thus the signalspassing through the gate circuit 110 correspond to the signals passingthrough the gate circuit 98 and provide an indication of a valuedesignated as YL in co-pending application Serial No. 217,478. Thesignals from the gate circuits 98 and 110 pass through the or network114 to the gate circuit 118.

The gate circuits 116 and 118 become opened for the passage of signalsonly after the left tube in the multivibrator 120 has been triggeredinto a state of non-conductivity by the rst pulse in the channel 16after the 22nd pulse position for each integrator. Since this pulseindicates that the pulses which follow in the channel 16 represent thenumerical value of the dependent quantity y for the integrator, the gatecircuits 116 and 118 pass information relating only to the numericalvalues of Ya and Ya, respectively.

The information passing through the gate circuits 116 and 118 isintroduced to the adder 126. The adder 1256 also receives informationfrom the gate circuits and 132, which pass the information appearing onthe plates of the left and right tubes in the multivibrator 56 onlyafter the 22nd pulse position for each integrator. Because of theconnection from the coil 24 to the grid of the left tube in themultivibrator 56, the plate of the left tube in the multivibrator 56indicates the integer l in the pulse positions in the channel 14 afterthe 22nd position for each integrator when the voltage on the plate ofthe tube is relatively high. Similarly, an indication of the integer 0is provided on the plate of the right tube in the multivibrator 56 forthe pulse positions after the 22nd pulse position for each integratorwhen the voltage on the plate of the tube is relatively high. As will bedisclosed in detail hereinafter, the information provided in the channel14 in the pulse positions after the 22nd pulse position for eachintegrator relates to the cumulative value of the differentialcombination yAx for the integrator. The indications of the integers land 0 in the channel 14 after the 22nd pulse position for eachintegrator are respectively designated as R and R in copendingapplication Serial No. 217,478.

The adder 126 combines the indications of R and R passing through thegate circuits 130 and 132, respectively, with the indications of Ya andYa' passing through the gate circuits 116 and 118, respectively and withthe carry information passing through the circuit 128. The circuit 128operates in a manner similar to the circuit 178 (Figure 3) to provide acarry of an integer from one pulse position to the next when a value of+2 or +3 is obtained by addition in the first pulse position. Forexample, when the values Ya and R are added together for the 26th pulseposition, the integer l is added to the integer 1 to obtain a fullbinary indication of +2. Since an indication of +2 is equivalent inbinary form to a value of 0 for the pulse position and a carry of +1 tothe next position, the circuit 128 (Figure l) carries a value of 1 tothe 27th position. This carry is arithmetically combined in the 27thpulse position with the indications passing through the gate circuits116, 118, 130 and 132. In this Way, a new Value is obtained for eachpulse position in the channel 14 after the 22nd pulse position for eachintegrator.

The operation of the adder 126 is similar to that disclosed inco-pending application Serial No. 217,478. As disclosed in theco-pending application and as shown in Figure 45 of the co-pendingapplication, the adder 176 prlvides a relatively high Voltage for apulse position W en In the above equation, D2 indicates a carry to onepulse position from the previous pulse position and D2 indicates theabsence of a carry. Furthermore, Q indicates an arithmetical combinationof the different quantities to produce a relatviely high voltagerepresenting the integer "1 for each pulse position. The sign indicatesan or proposition such that Q is true when any one of the andpropositions YaRl'Dz, Ya'RlDZ, Ya'RlDZ or YaRlDz is true.

Similarly, the integer 0 is obtained for a pulse position when In theabove equation, Q indicates the integer 0 since it is the inverse of Q.

The output from the adder 126 is introduced to the gate circuit 138(Figure 2). Because of its connection through the line 106 to thecounter 52, the gate circuit 138 is prepared for opening only after the22nd pulse position for each integrator. The gate circuit becomes openfor the passage of signals in these pulse positions only when relativelyhigh voltages are simultaneously introduced to it from the adder 126 andthe plate of the 1eft tube inv the multivibrator 140.

Because of the connection to the grid of the left tube in themultivibrator 146 from the gate circuit 141, the left tube in themultivibrator 140 becomes cut off only when an actual Ax incrementoccurs for an integrator. The left tube in the multivibrator 140 becomescut olf only when an actual Ax increment occurs, since the gate circuit141 opens and passes a signal only upon the occurrence of an actual Axincrement.

Since the gate circuit 141 is connected through the line 83 to thecounterv 52, it becomes prepared for opening only during the first 22pulse positions of each integrator. The gate circuit also receives thepulses of relatively high voltage which are induced in the coil 24,since the gate circuit is connected to the plate of the left tube in themultivibrator 56.` Such pulses include a coded indication in the iirst22 pulse positions for each integrator that a Ax increment may actuallyoccur for the integrator.

The gate circuit 141 also receives the pulses of relatively high voltageappearing on the plate of the left tube in the multivibrator 60. Aspreviously disclosed, the left tube in the multivibrator 60 is cut offin a pattern dependent upon the magnetic pulses appearing in the channel18 and induced as electrical pulses in the coil 40. Each of the pulsesin the channel 18 provides an indication of an actual Ax increment foran integrator when it coincides with a pulse appearing in the channel 14in one of the rst 22 pulse positions for the integrator. Since the gatecircuit 141 opens upon the simultaneous occurrence of pulses in thechannels 14 and 18 during the iirst 22 pulse positions for eachintegrator, a signal passing through the gate circuit 141 for anintegrator provides an indication that a Ax increment has actuallyoccurred for the integrator.

Upon the passage of a signal through the gate circuit 141, the left tubein the multivibrator 140 becomes cut H so as to prepare the gate circuit133 for opening. The gate circuit 138 opens after the 22nd pulseposition for each integrator and passes the information produced by theadder 126. This information passes through the Lor network 142 and theor network 84 for recordation by the coil 28 in the channel 14. In thisWay, a new indication is obtained for the cumulative value of the yAxincrements for each integrator when a Ax increment actually occurs forthe integrator.

After the gate circuit 138 has been opened for the passage ofinformation for a particular integrator, it remains lopen until the 48thpulse position for the integrator is presented for utilization. Upon thepresentation of the 48th pulse position for the integrator, the gatecircuit 13S closes because of its connection through the line 1% to thecounter 52. After the gate circuit 138 has closed, the right tube in themultivibrator 140 is triggered into a state of non-conductivity as aresult of a signal introduced to the grid of the tube through the line97 from the counter 52.

When the right tube in the multivibrator 1.40 becomes cu-t off, the lefttube starts to conduct and causes a relatively low voltage to beproduced on its plate. This low voltage prevents any information frompassing through the gate circuit 138 for the next integrator to bepresented for computation unless an actual Ax increment is obtained forthat integrator. If an actual Ax increment occurs for the integrator, asignal passes through the gate circuit 141 in a manner similar to thatdisclosed above and triggers the left tube in the multivibrator 140 intoa state of non-conductivity.

As may be seen in Figure 5, a differential combination of yAx should beobtained only when an actual Ax increment occurs. For example, a yAxincrement indicated by the shaded areav 2.0.0 in Figure 5 is obtainedwhen an actual Ax increment indicated at 240 occurs. If an actual Axincrement does. not occur for an integrator upon the presentation of theintegrator for computation, the cumu- 16 lative value of the yAxincrements for the integrator should remain unchanged. This isaccomplished in the analyzer constituting this invention by closing thegate circuit 138 (Figure 2) andv opening the gate circuit 143 when a Axincrement does not actually occur for an integrator.

As previously disclosed, the plate of the left tube in the multivibrator141i continues to conduct when an actual Ax increment does not occur foran integrator. The resultant relatively low voltage on the plate of theleft tube in the multivibrator 1.40 prevents the gate circuit 138 fromopening to pass the new information obtained by the adder 126. Since theright tube in the multivibrator is cut off during the time that the lefttube is conducting, a relatively high voltage is produced on the plateof the right multivibrator tube. introduced to the gate circuit 143 topreparethe gate circuit for opening.

Because of the connection to the gate circuit 143 through the line 106from the counter 52, the gate circuit is able to open only after the22nd pulse position for each integrator. In this way, the pulses ofrelatively high voltage appearing on the plate of the left tube in themultivibrator 56 are able to pass through the gate circuit 143 after the22nd pulse position for eachV integrator. Since voltage pulses areproduced on the plate of the left tube in the multivibrator 56 in apattern corresponding to that induced in the coil 24, the pulsesappearing in the channel 14 are introduced to the gate circuit 143. Thiscausesfthe pulses appearing in the channel 14 to pass through a circuitincluding the coil 24, the multivibrator 56, the gate circuit 143, theor networks 142 and 84 and the coil 28 after the 22nd pulse position foran integrator when a Ax increment has not actually been obtained for theintegrator. The pulses introduced to the coil 28 are recorded in thechannel 14 and are subsequently presented again to the coil 24 forutilization.

The output from the carry circuit 128 is introduced to the gate circuit146. Because of its connection through the line 97 to the counter 52,the gate circuit is prepared for opening only during the 48th pulseposition of each integrator. In this way, the gate circuit is able to.pass only a carry indication which is obtained by the arithmeticalcombination of the dilierent values in the 47th pulse position for eachintegrator and which is carried to the 48th pulse position for theintegrator. Since the gate circuit is connected to the plate of the lefttube in the multivibrator 140, the gate circuit can open in pulseposition 48 for an integrator only when a Ax increment has actuallyoccurred for the integrator.

The gate circuit 146 is also connected to the output terminal of the ornetwork 112. As a result of this connection, a carry signal from thecircuit 12S is able to pass through the gate circuit 146 in the 48thpulse position for an integrator only when the value of Ya is positive.When Ya is positive for an integrator, it is added directly into thevalue of the cumulative yAx increments provided in the channel 14 forthe integrator. In this way, onlya positive overflow can be obtained inthe cumulative yAx value for the integrator, and this overiiow isindicated by a pulse from the carry circuit 128.

The operation of the gate circuit 146 may be seen more clearly byreference to a particular example. For example, if a positive decimalnumber such as 596 is indicated inl binary form in the channel 14 for aparticular integrator and if the maximum indicationthat can be providedis 600, the addition of a value of +6 representing the value of Yacauses the indication in the channel 14 to return to a decimal value of+2. At the same time, the adder 126 produces a pulse for carry from the47th position to the 48th position for the particular integrator, andthis carry pulsev passes through the circuit 128. Since the value of Yais positive, the carry pulse then passes through the gate circuit 146 toindicate an overilow .in the cumulative yAx value provided in thechannel 14 for the integrator.

This voltage is Because of the binary indications in the channel 14,Y`

a full indication in the cumulative yAx value for an integrator isprovided by a pulse in the channel 14 in each of the pulse positionsrepresenting the cumulative value of yAx for the integrator. As aresult, an overow in the cumulativeyAx value for an integrator isobtained only after a pulse of relatively high intensity simultaneouslyappears in each pulse position representing the cumulative value of yAxfor the integrator. Upon an overflow, the indications in the channel 14for the integrator return to a relatively low intensity for each pulseposition to indicate the integer for each pulse position. Subsequentadditions of the positive Ya indications for an integrator cause theindications in the channel 14 for the integrator to increase in value.

The output from the gate circuit 146 is applied through the or networks154 and 90 to the coil 38 and is recorded by the coil 38 in the channel18. Since the gate circuit 146 passes only a signal for each integratorto indicate that an overow has occurred in the cumulative yAx valuestored in the channel 14 for the integrator, the channel 18 indicates indifferent pulse positions Whether or not an overow has occurred for eachintegrator the last time thatv the integrator was presented forcomputation.I The operation of the channel 18 to present the overflowinformation for the different integrators will be disclosed in detailhereinafter.'

The output from the gate circuit 146 is also introduced to the gatecircuit 158. The gate circuit 158 also has a Voltage introduced to itfrom the plate of the right tube in the 4multivibrator 56. This causesthe signal from the gate circuit 146 to pass through the gate circuit158 only when a relatively high voltage is simultaneously introduced tothe gate circuit 158 from the plate of the right tube in themultivibrator 56. As previously disclosed, the voltage on the plate ofthe right tube in the multivibrator becomes relatively high fordifferent pulse positions to indicate'the value 0 for the positions.

As disclosed in co-pending application Serial No. 217,- 478, the 48thpulse position for each integrator in the channel 14 is utilized toprovide a code as to whether or not the cumulative yAx value stored inthe channel 14 for Ithe integrator is to be inverted in polarity. Thiscode is provided in the channel 14 in pulse position 48 for eachintegrator at the beginning of a problem and is recirculated through thegate circuit 76 and the or network 84 during the actual computation. Therecirculation occurs because of the connection to the gate circuit 76from the counter 52 through the line 97. In this way, the codedinformation in pulse position 48 for each integrator is made availableto the integrator every time that the in-V tegrator is presented forcomputation during the actual solution of a problem. n

Inco-pending application Serial No. 217,478, a pulse is provided in thechannel 14 in pulse position 48 for an integrator when the cumulativeyAx value stored in the channel 14 for the integrator is to be invertedin polarity.

a p'ulse in the channel 14 in pulse position 48 for thev integrator. Aspreviously disclosed, the gate circuit 146 passes a carry pulse from thecircuit 128 only whena carry occurs upon the addition of positive YELvalues into theadder 126. Since the .Ya value is positive and since theR value storedY in the channel 14 isto-be maintained `in I the samepolarity, the gate circuit 158-passes a signal to indicatekthatthegoverowffor. an integrator is positive.

'Ihesignaisgpassnglthroush the gatecircuit, 15.8 pass through the ornetworks 162 and 92 to the coil`l44'foy recordation by the coil 44 inthe channel 20. ln this way, the pulses appearing in the channel 20provide an indication of a positive overow in the cumulative yAx valuefor an integrator when a pulse appears at the same time in the channel18 to indicate that an overow has actually occurred.

The output from the carry circuit 128 is also introduced to the gatecircuit 152. The output is introduced to the gate circuit 152 through aline which causes the carry indications from the circuit 128 to appearas pulses of relatively low voltage and the absence of carry indicationsto appear as pulses of relatively high voltage. Because -of theinversion of the output from the gate circuit 128, the gate circuit 152can open only when a relatively low ouput indicative of the integer 0 isprovided by the carry circuit 128.

The gate circuit152 can open only at the 48th pulse` position'forV eachintegrator since it is connected through the line 97 to the counter 52.Furthermore, the gate circuit 152 can open at the 48th pulse positionfor an in?l integrator yonly when the value Ya appears fortheintegrator. When the gate circuit 152 opens, a signal passes throughit and through the or networks 154 and 90 to the coil 38 for recordationby the coil 38 in the `channel 18. The passage of a signal through thegate circuit 152 provides an indication that an overow has occurred inthe cumulative yAx value stored for the integrator in the channel 14. Inthis respect, the gate circuit 152 operates in a manner similar to thegate circuit 146.

As previously disclosed, the term Ya' is indicated by the equaltionYa=(-Y) (AxH-(Y) (-Ax). As may be seen by either the expression (Y)(-Ax) or the expression (-Y) (Ax), the term Ya' for an integratorindicates that a negative number is being introduced to the adder 126for arithmetical combination with the cumulative yAx value sto-red inthe channel 14 for the integrator. Since Ya' is negative, only anegative overflow can occur in the cumulative yAx value for theintegrator when the value of Ya" is arithmetically combined with thecumulative yAxV value for the integrator.

As previously disclosed, a full indication of a positive numberrepresenting the cumulative yAx value for an integrator is obtained by apulse of relatively high voltage for each pulse position in the channel14 for the in-y Since a negative overow in the channel 14` is obtainedfor an integrator when the integrator has a negative value representedby Ya', the o verow can occur only after-a pulse of low voltage hasappeared in each informationv kposition in the channel 14 for theintegrator upon the l presentation of the integrator Vfor computation.-A Upon a.

negative overflow in the channel 14for an integrator, they indicationsin the channel 14 for the integrator return to a relatively highintensity -f0r each pulse position to indicate the integer 1. for eachposition. Subsequent additions of the negative Ya' indications for anintegrator causethe indications in the channel 14 for the integrator toincrease in magnitude. An increase inmagnitude in a` negative directionis obtained Aby progressively changing 19 tionsof low intensity forpulse positions of increasing numerical signiticance.

, For example, a decimal value of -596 may be indicated in binary :formin the channel 14 for a particular integrator. If the maximum negativeindication that can be provided as the cumulative yAx value for theintegrator is 600, the arithmetical combination of the value of 96 and avalue of -5 representing the Ya' value for the integrator causes anegative overow to be obtained. At the same time, the circuit 128provides a carry from the 47th pulse position to the 48th pulse positionfor the particular integrator to indicate that a negative overflow hasoccurred in the cumulative yAx value for the integrator. Since theoverow is negative, the circuit 128 provides an indication of the overowby a pulse of relatively low intensity at the 48th pulse position. Thiscorresponds to the value D2' for pulse position 48 of the integrator.

A signal passing through the gate circuit 152 at pulse position 48 foran integrator provides an indication that an overflow has occurred. T hepolarity of such an overflow for an integrator is determined by thepresence or absence of a coding pulse in the channel 14 in pulseposition 48 for the integrator. The presence of a coding pulse in pulseposition 48 for an integrator indicates that the cumulative yAx valuestored in the channel 14 for the integrator is to be inverted inpolarity. The presence of such a coding pulse is indicated by arelatively high voltage on the plate of the left tube in themultivibrator 56 at pulse position 48 for the integrator.

When a pulse is produced by the gate circuit 152 at pulse position 48for an integrator and a pulse of relatively high voltage issimultaneously produced on the plate of the left tube in themultivibrator 56, a signal passes through the gate circuit 160. Thissignal indicates that the pulse passing through the gate circuit 152represents a positive overow in the cumulative yAx value for theintegrator. A positive overflow is obtained because the coding pulse inthe channel 14 in pulse position 48 for the integrator causes thepolarity of the overflow pulse passing through the gate circuit 152 tobe inverted. Since the pulse passing through the gate circuit 152indicates a negative overflow, the inversion of such a negative overowby the gate circuit 160 causes the gate circuit `to indicate a positiveoverow.

When a pulse passes through the gate circuit 160 to indicate a positiveoverow for an integrator, it is introducedvto the coil 44 after passingthrough the or networks 162 and 92. The pulse is then recorded in thechannel to indicate that the overlowpulse simultaneously recorded in thechannel 18 is positive. In this way, the gate circuit 160 operates in amanner similar to the gate circuit 158 to indicate a positive overflow.

The operation of the gate circuit 146 may be indicated by the logicalequation Zt=TD2YaP48. In the above equation, T indicates a relativelyhigh voltage on the plate of the left tube in the multivibrator 140 andP48 indicates pulse position 48 of each integrator. Zt indicates that amagnetic pulse of relatively high intensity is recorded in the channel18V. Similarly, the operation of the gate circuit 152 is indicated bythe logical equation Zt=TD2"Ya'P4g. v Y

The logical equation for the operation of the gate circuit 158 may beindicated as Z =(TD2YaP48)(R1), where 7s indicates that a magnetic pulseof relatively high intensity is recorded in the channel Z0. Similarly,theA logical equation for the operation of the gate circuit 160 isindicated by the equation Z =(TD2Ya'P48)(R1).

As previously disclosed, the coils 38 and 44 are eiectively separated by49 pulse positions from the coils 40 and V46, respectively. Since thelength of each integrator is only 48 positions, a precessing actionoccurs in the channels 18 and 20. This precessing action causes a pulseposition to be made available inleach of the channelsr18 and 20 so thatthe overow information for the cumulative yAx value in the 48th pulseposition yfor each 2G integrator can be recorded after the computationhas been made for the integrator. shown in Figure 8.

In all of the vertical columns in the chart shown in Figure 8 except forthe two at the extreme right, numbersV between l and "22 are showncorresponding to the 22 integrators in the digital differential analzer.In the two vertical columns at the extreme right, numbers are shownprefaced by the letters I and P. The letter I followed by a numberindicates the particular integrator that is moving past the coils 38 and44 at any instant. For example, I3 indicates that a pulse position inthe third integrator is moving past the coils 38 and 44. Similarly, adesignation such as P13 indicates that the 13th pulse position in theparticular integrator is moving past the coils 38 and 44.

Since the operation of the channel 20 is similar to that of the channel18, the following discussion will be limited to the channel 18. As willbe seen at 250 in Figure 8, a iirst pulse of relatively high voltage maybe provided in the channel 18 at the 48th position of Integrator 1. Thispulse indicates that an overilow has occurred in the cumulative yAxvalue stored in the channel 14 for the integrator but the pulse does notindicate whether the overfiowV is positive or negative. The pulse 250advances from the coil 38 towards the coil 40 as the drum 10 rotatesthrough the 48 positions of Integrator 2. At the 48th position ofIntegrator 2, a pulse of relatively high voltage may be recorded by thecoil 38 to indicate an overow from Integrator 2, as shown at 252 inFigure 8. At the P113 position, the indication 250 passes through thegate circuit and the or network 90 to the coil 38. This pulse is againrecorded by the coil 38 in the channel 18, this time at the pulseposition adjacent to the indication 252.

Similarly, indications are provided in adjacent pulse positions to showwhether or not an overflow has occurred in the cumulative yAx value foreach of the other integrators in the analyzer. These indications arerecirculated by the gate circuit 80, which remains open during the iirst47 puise positions of each integrator. At the 48th position for eachintegrator, the gate circuit 80 closes and prevents any recirculation ofoldA infomation. At the same time as the gate circuit 80 closes, theoverf ow information for the integrator moving past the coil 38 isrecorded in the channel 18. In this way, old overow information for anintegrator is replaced by new overow information for the integratorevery time that the integrator is presented for computation.

After the indications have been provided in the channel 18 for the 48thpulse position of each integrator, Integrator l becomes available forcomputation a sec ond time. sitions for the integrator, the outputindications for the 22integrators move in sequence past the coil 40.

' digital differential analyzer to obtain such a determination has beenpreviously disclosed in detail.

VIn like manner, the pulse indications inthechannel 18 representing theoveriiow indications for the different integrators are made available toeach integrator as it is' presented for computation. The overflowindications in the channel 20 are also made available to each integratorlas it is presented for computation so that a determination` can be madeas to whether each actual Ax increment `is `positive or negative and sothat asimilar determina tion can be made-as to the polarityuof each Ayincrement. The digital differential analyzer disclosed vabove hasseveral. important advantages. By determining-whether This may be seenin the chart-y As the drum rotates through the iirst 22 po-y This,`causes the output indications in the channel 18 to become is obtainedfor the cumulative yAx value for each inte.

grator every time that the integrator is presented for computation.Because of this ternary indication, the digital differential analyzer isable to provide a determination on a ternary basis as to whether each Axincrement is -l-l, or l. A similar ternary indication is also providedby the analyzer for each Ay increment. Since Ax and Ay *increments foran integrator can be determined to be 0 as well as -l-l and -1, truevalues of the independent quantity x and the dependent quantity y foreach integrator can be obtainedat all times. This causes the accuracy ofthe digital differential analyzer disclosed above to be enhanced overthat obtained from ianalyzers now in use. v

It should be appreciated that a systemV of ternary transfer similar tothat disclosed above can be incorporated in other digital differentialanalyzers than that disclosed in copending application Serial No.217,478. For example, the system of ternary transfer disclosed above canbe easily adapted for use with the digital differential analyzerdisclosed in co-plending application Serial No. 263,152, led December26, 1951 by Glenn E. .Hagen et al.

It should be Yappreciated that the term signal indications as used inthe claims refers to physical phenomena which take place in the digitaldifferential analyzer, as forV example, the production of electricalsignals. The termV differential combination as used in the claims refersto the combination of the dependent quantity y and the increments Ax inthe independent quantity to obtain the output quantity. represented bythe yAx increments.

vAlthough this invention has been disclosed and illustrated withreference to particular applications, the principles involved aresusceptible of numerous other applications which will be apparent topersons skilled in the art. The invention is, therefore, to be limitedonly as indicated by the scope of the appended claims.

. What is claimed is:

l. AA digital vdifferential analyzer, including, a plurality ofintegrator storage sections, means for providing digital signalindications alternativelyY representing: a positive increment, anegative increment, or no Yincrement in an independent quantity fromeach integrator storage section, means for providing digital signalindications representing a dependent quantity from each integratorstorage section, and means for combining the signal indicationsrepresenting the dependent quantity and variations in the independentquantity from each integrator storage section upon the occurrence ofdigital variations in the independent quantity for the integratorstorage section and in accordance with the`polarity of such occurrencesto provide signal indications representing the differentiall-dependentquantity, means for providing digital signal indications.representing a differential combination from each integrator storagesection, and means for combining the signal indications representing thedependent quantity and the differential combination from an integratorstorage section upon the transfer of the signal indicavtionsrepresenting the dependent quantity from said transfer means, to therebyprovide signal indications digitally representing a new value of thedifferential combina-v tion.

3. A digital differential analyzer, including a plurality of integratorstorage sections, means for providing digital signal indications capableof representing: a discrete positive variation, a discrete negativevariation, orV no variation in an independent quantity from anintegrator storage section, means for providing signal indicationsdigitally representing adependent quantity from an integrator storagesection, means for providing for the transfer of the signal indicationsrepresenting the dependent quantity upon the occurrence of digitalvariations 'in the' independent quantity, means for providing digitalsignal indications representing the cumulative value of a differentialcombination from an integrator storage section, means for combining thesignal indications representing the dependent quantity and thedifferential combination upon transfer of the signal indicationsrepresenting the dependent quantity from an integrator storage sectionto thereby produce signal indications digitally representing a newcumulative valuevof the differential combination, and means forregistering the signal indications, representing the cumulative value ofthe differential combination.

4. A digital differential analyzer, including, a plurality of integratorstorage sections, electrical circuitry for providing digital signalindications capable of representing: a discrete positive Variation, adiscrete negative variation,

. or no variation in an independent quantity from each integratorstorage section, electrical circuitry for providing digital signalindications representing the polarity` of variationslin an independentquantity from each integrator storage section, electrical circuitry forproviding digital signal indications representing a dependent quantityfrom each integrator storage section, electrical circuitry fortransferring the digital signal indications representing the dependentquantity from each integrator storage secthe differential combination,and electrical circuitry forv registering the signal indicationsrepresenting the differential combination in an integrator storagesection.

5. A differential analyzer comprising: a recording means; means fordefining said recording meansfinto a plurality of'integratory storagesections; lsensing meansy for sensing said recording means; means forscanning said recording means with said sensing means to thereby:

derive a plurality of digital signal indications from each of saidintegrator .storage sections including: aA signal indicationsalternatively representing a positive increment, `a negative incrementor no increment, in an independent quantity, signal indicationsrepresenting a dependent quantity, and signal indications representingan accumulated value ofthe different combination-lofincrements in saidindependent quantity and said dependent vquantity; means fordifferentially combining said lsignal indications representing anincrementin anv independent quantityand said signal indicationsrepresenting a dependent quantity to form. new differential combi-lnation signal indications; and means forV algebraically combining saidnew differential Y combination signal indications and said signalindications representing an accumulated value of the differentialcombinations.

6. A differential analyzer comprising; a recording means; means fordefningfsaid recording means into a anodis plurality of integratorstorage sections; sensing means for sensing said recording means; meansfor scanning said recording means with said sensing means to therebyderive a plurality of digital signal indications from each of saidintegrator storage sections including: signal indications alternativelyrepresenting a positive increment, a negative increment, or no incrementin an independent quantity, signal indications alternativelyrepresenting an increment or no increment in a dependent quantity,signal indications representing an accumulated dependent quantity, andsignal indications representing an accumulated value of the diierentialcombination of increments in said independent quantity and said de-Vpendent quantity; means for algebraically combining said increments in adependent quantity and said accumulated dependent quantity to formsignal indications representing a new dependent quantity; means fordifferentially combining said signal indications representing anincrement in an independent quantity and said signal indicationsrepresenting a new dependent quantity to form new differentialcombination signal indications; and means for algebraically combiningsaid new differential combination signal indications and said signalindications representing an accumulated value of the differentialcombinations.

7. A differential analyzer comprising: a recording means; means fordefining said recording means into a plurality of integrator storagesections; sensing means for sensing said recording means; means forscanning said recording means with said sensing means to thereby derivea plurality of digital signal indications from each of said integratorstorage sections including: signal indications alternativelyrepresenting a positive increment, a negative increment, or no incrementin an independent quantity, signal indications representing a dependentquantity, and signal indications representing an accumulated value ofthe differential combination of increments in said independent quantityand said dependent quantity; means for differentially combining saidsignal indications representing an increment in an independent quantityand said signal indications representing a dependent quantity to formnew differential combination signal indications; means for algebraicallycombining said new diiferential combination signal indications and saidsignal indications representing an accumulated value of the differentialcombinations to thereby form signal indications representing a newaccumulated value of the differential combinations; and means forre-registering said signal indications representing a dependent quantityand said signal indications representing an accumulated value of thedifferential combination in said recording means.

8. A di'terential analyzer comprising: recording means for recordingdigital signals; means for deiining said recording means into aplurality of integrator storage sections; sensing means for sensing saidrecording means; means for scanning said recording means with saidsensing means to thereby derive a plurality of digital signalindications from each of said integrator storage sections including:signal indications alternatively representing a positive increment, anegative increment or no increment in an independent quantity, signalindications representing a dependent quantity, and signal indicationsrepresenting an accumulated value of the differential combination ofincrements in said independent quantity and said dependent quantity;means for differentially combining said signal indications representingan increment in an independent quantity and said singal indicationsrepresenting a dependent quanti'ty to form new differential combinationsignal indications; and means for algebraically combining said newditferential combination signal indications and said signal 24indications representing an accumulated value of the differentialcombinations.

9'. A digital differential analyzer including: a plurality.

of integrator storage sections; means for providing digital signalindications from each of said integrator storage sections representing adependent quantity; means for providing digital signal indications fromeach of said integrator storage sections representing an accumulateddierential combination of a dependent quantity and increments in anindependent quantity; means for providing digital signal indicationsfrom each of said integrator storage sections capable of representingpositive increments, negative increments and no increment ,in

an independent quantity; means for differentially combining said signalindications representing a dependent quantity and said increments in anindependent quantity, to form signals representing an incrementaldilerentialcombination; and means for algebraically combining saidsignals representing an incremental differential combination and saidsignals representing an accumulated differential combination.

l0. A digital differential analyzer including: a plurality of integratorstorage sections; means for providing digital signal indications fromeach of said integrator storage sections Yrepresenting a dependentquantity; means for providing digital signal indications from each ofsaid integrator storage sections representing variations and novariation in a dependent quantity; means for altering said signalindications representing a dependent quantity according to said signalindications representing variations in a dependent quantity; means forproviding digital signal indications from each of said integratorstorage sections representing an accumulated differential combination ofa dependent quantity and increments in an independent quantity;means'for providing digital signal indications from each of saidintegrator storage sections capable of representing positive increments,negative increments and no increment in an independent quantity; meansfor dierentially combining said signal indications representing adependent quantity and said increments in an independent quantity, toform signals representing an incremental dilferential combination; andmeans for algebraically combining said signals representing anincremental dilerential combination and said signals representing anaccumulated differential combination.

References Cited in the le of this patent UNITED STATES PATENTS OTHERREFERENCES A New Type of Differential Analyzer, by V. Bush et al.:Journal of the Franklin Institute, volume 240, No. 4, October 1945,pages Z55-326.

Fundamental Concepts of the Digital `Differential Analyzer Method ofComputation, R. E. Sprague, Mathe matical Tables and Other Aids toComputation, volume 6, January 1952, pages 41-49 only.

The Serial-Memory Digital Differential Analyzer,

I. F. Donan, Mathematical Tables and Other Aids to- Computation, volume6, April 1952; pages 102-112 only.

